Welcome to We The Armed cpolk!
Often your load book will mention a specific primer that was used in working up the data. For example this data from Alliant
mentions the use of WSR primers - that is W
Generally though, you can pretty easily infer the size for yourself. For most metallic cartridges using boxer primers there are only four sizes to choose from: small pistol, large pistol, small rifle, and large rifle. Look at a few different cartridges and you'll readily see which ones are large and which are small.
The small pistol and small rifle are the same diameter, as are the large pistol and large rifle. Pistol and revolver cartridge usually use pistol primers and rifle cartridges generally use rifle primers.
The remaining factor is magnum primers. Any of the four sizes may be had in a magnum variation and these primers generate a larger/hotter flame for igniting heavier loads of slower burning powders and IIRC may use a thicker cup to withstand higher pressures. Again, this is something that should be mentioned in your load books if it's necessary.
From your example; 45/70 gov't will use large rifle, .223 Remington will use small rifle, .308 Winchester will use large rifle, and .44 Mag will use large pistol.